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Solve the following nonlinear system algebraically. be sure to check for non-real solutions.

Solve the following nonlinear system algebraically. be sure to check for non-real-example-1

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\begin{gathered} \text{Given} \\ x^2+y^2=8\text{ \lparen first equation\rparen} \\ 2x^2+4y^2=34\text{ \lparen second equation\rparen} \end{gathered}

Use substitution method to and solve in terms of x using the first equation


\begin{gathered} x^2+y^2=8 \\ x^2=8-y^2 \end{gathered}

Next, substitute it to the second equation


\begin{gathered} 2x^2+4y^2=34 \\ 2(8-y^2)+4y^2=34 \\ 16-2y^2+4y^2=34 \\ 2y^2=34-16 \\ 2y^2=18 \\ (2y^2)/(2)=(18)/(2) \\ y^2=9 \end{gathered}

Then, substitute it back to first equation and solve for x


\begin{gathered} x^2+y^2=8 \\ x^2+9=8 \\ x^2=8-9 \\ x^2=-1 \end{gathered}

Now we have the following solutions


x^2=-1\text{ and }y^2=9

Get the square root of both x's and y's to get the solution


\begin{gathered} x^2=-1 \\ √(x^2)=√(-1) \\ x=\pm i \\ x=i\text{ and }x=-i \\ \\ y^(2)=9 \\ √(y^2)=√(9) \\ y=\operatorname{\pm}3 \\ y=3\text{ and }y=-3 \end{gathered}

Getting the combination of ordered pairs we have the following solutions


\begin{gathered} (i,3) \\ (i,-3) \\ (-i,3) \\ (-i,-3) \end{gathered}

User Ali Sarchami
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